On the odd-minor variant of Hadwiger's conjecture

Authors:
Jim Geelen;Bert Gerards;Bruce Reed;Paul Seymour;Adrian Vetta
Affiliations:
Department of Combinatorics and Optimization, University of Waterloo, Canada;Centrum voor Wiskunde en Informatica, and Eindhoven University of Technology, Netherlands;School of Computer Science, McGill University, Canada;Department of Mathematics, Princeton University, USA;Department of Mathematics and Statistics, and School of Computer Science, McGill University, Canada
Venue:
Journal of Combinatorial Theory Series B
Year:
2009

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Abstract

A K"l-expansion consists of l vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains no odd K"l-expansion then its chromatic number is O(llogl). In doing so, we obtain a characterization of graphs which contain no odd K"l-expansion which is of independent interest. We also prove that given a graph and a subset S of its vertex set, either there are k vertex-disjoint odd paths with endpoints in S, or there is a set X of at most 2k-2 vertices such that every odd path with both ends in S contains a vertex in X. Finally, we discuss the algorithmic implications of these results.